Higher GCSE

# How to find the nth term of a quadratic sequence

Following on from our previous blogs on identifying different types of sequences and finding the nth term of a linear sequence, in this blog I will show you how to find the nth term of a quadratic sequence. It is important to note that this topic is only examinable on the higher tier GCSE maths exam.

Let’s work through an example.

Example 1
Find the nth term of the quadratic sequence 3, 4, 7, 12, …

Before we start, because this is a quadratic sequence, we know our nth term formula is going to be of the form an2 + bn + c. We just have to find a, b and c.

First, let’s find a. To find a, we find the difference of the differences in our sequence, and then divide this by 2.

The differences of the differences are in green below:

This gives us 2.

Now, divide this number by 2:

2 ÷ 2 = 1

We have found a (in other words, the coefficient of n2 in our nth term formula).

Now, because we know our nth term contains 1n2 (or just n2), we are going to write out our sequence, compare it to the sequence n2 and then find the difference between these two. This difference will produce a sequence which we can use to find the rest of our nth term formula.

Below, in black, is our sequence.
Then, in red, is the sequence n2.
Then, in green, is the difference between our sequence and the sequence n2.

The differences between our sequence and the sequence n2 now forms a linear sequence (in green above). This sequence should always be linear – if it isn’t, you have done something wrong. What we now need to do is find the nth term of this green sequence. We will need to add this on to n2 – this will tell us our b and c. If you need a reminder of how to find the nth term of a linear sequence, you can re-read the previous blog.

The sequence has a difference of -2, and if there were a previous term it would be 4.

So the nth term of the green sequence is -2n + 4.

Adding this on to what we already knew, this means our nth term formula is n2 – 2n + 4.

What I would strongly recommend at this stage is that you check your answer. Going back to why the nth term formula is useful, remember that the formula tells you any term in the sequence. We know from the question that the first term in the sequence is 3. So, if we plug 1 into the formula we should get 3. Likewise, we know that the second term in the sequence is 4, so if we plug 2 into the formula we should get 4. This allows us to check the formula we calculated is correct. And if we plug in 3, we should get 7.

n = 1                      12 – 2×1 + 4 = 1 – 2 + 4 = 3                              this matches our sequence!
n = 2                      22 – 2×2 + 4 = 4 – 4 + 4 = 4                              this also matches!
n = 3                      32 – 2×3 + 4 = 9 – 6 + 4 = 7                              this also matches!

Let’s do the fourth term as well, we know this should be 12…

n = 4                      42 – 2×4 + 4 = 16 – 8 + 4 = 12                         4 out of 4!

So we are sure our answer is correct. It’s always a nice feeling, not just in maths, when you give an answer and you know it is correct. I would recommend always try at least 2 terms, because you could always fluke one!

Example 2
Find the nth term of the quadratic sequence 1, 3, 9, 19, …

First, find a – the difference of the differences divided by 2.

The difference of the differences is 4 this time, so 4 ÷ 2 = 2, giving us a = 2.

So we know our sequence starts with 2n2.

Now, compare our sequence with the sequence 2n2 (this is just the sequence for n2 but each term multiplied by 2).

This generates the linear sequence in green. We now need to find the nth term of this sequence.

The sequence has a difference of -4 and if there were a previous term it would be 3.

So the nth term of the green sequence is -4n + 3.

So our final nth term formula will be 2n2 – 4n + 3.

Now, let’s check the first three terms…

n = 1                      2×12 – 4×1 + 3 = 2 – 4 + 3 = 1                         this matches our sequence
n = 2                      2×22 – 4×2 + 3 = 8 – 8 + 3 = 3                         this also matches
n = 3                      2×32 – 4×3 + 3 = 18 – 12 + 3 = 9                    this also matches

So we are confident our answer is correct.

Example 3
Find the nth term of the quadratic sequence 2, 3, 10, 23, …

First, find a.

6 ÷ 2 = 3.

So the nth term begins with 3n2. Now compare our sequence to this.

Now find the nth term of the green sequence.

The sequence has a difference of -8 and if there were a previous term it would be 7.

So the nth term is -8n + 7.

Giving our final answer as 3n2 – 8n + 7.

Check the first three terms…

n = 1                      3×12 – 8×1 + 7 = 3 – 8 + 7 = 2                         this matches our sequence
n = 2                      3×22 – 8×2 + 7 = 12 – 16 + 7 = 3                    this also matches
n = 3                      3×32 – 8×3 + 7 = 27 – 24 + 7 = 10                  this also matches

Example 4
Find the nth term of the quadratic sequence 8, 13, 20, 29, …

First, find a.

2 ÷ 2 = 1.

So the sequence begins with n2. Now compare our sequence to this.

Now find the nth term of the green sequence.

The sequence has a difference of 2 and if there were a previous term it would be 5.

So the nth term is 2n + 5.

Giving our final answer as n2 + 2n + 5.

Check the first three terms…

n = 1                      12 + 2×1 + 5 = 1 + 2 + 5 = 8                            this matches our sequence
n = 2                      22 + 2×2 + 5 = 4 + 4 + 5 = 13                          this also matches
n = 3                      32 + 2×3 + 5 = 9 + 6 + 5 = 20                          this also matches

So there is the method for finding the nth term of a quadratic sequence. It’s not the easiest of methods, but with some practice you can pick it up fairly quickly.

Try finding the nth term of these 5 quadratic sequences. Put your answers in the comments or email your answers to sam@metatutor.co.uk and I’ll let you know if you go them right. If you need someone to explain this to you in person, book in a free taster session.

1. 1, 7, 15, 25, …
2. 7, 12, 21, 34, …
3. 5, 12, 25, 44, …
4. 0, 9, 22, 39, …
5. 10, 19, 34, 55, …

How to fix a calculator that is in the wrong mode
If you’re studying GCSE or A Level maths, you will be spending …
Perfect numbers
“If you look for perfection, you’ll never be content”There are very few …
The Capture-Recapture Method – how to estimate the number of fish in a lake
In the previous blog, I explained the Difference of Two Squares, as …
The Difference of Two Squares
Mentioned specifically in the Edexcel Advance Information for the 2022 GCSE exams …
Advance Information for Summer 2022 GCSE Exams
Due to the disruption to schooling caused by the Coronavirus pandemic, exam …
Times Tables App Review – Math Ninja
For a child in primary school, there is nothing more important in …
How to use triangles to remember exact sin, cos and tan values
If you’re taking the higher maths GCSE, there are certain values of …
A more efficient method for listing the factors of a number
In this blog I will show you the most efficient way to …
GCSE and A-Level Results 2021
Unfortunately due to the pandemic, GCSE and A-Level exams were cancelled again …
The Look-and-Say Sequence
In a previous blog I looked in more detail into “the daddy …
How to represent a recurring decimal as a fraction
What is a recurring decimal? Some decimal numbers are easy to deal …
Divisibility rules for numbers 7-12
Following on from the previous blog, which showed you a series of …
Divisibility rules for numbers 2-6
In this blog I will show you how you can easily check …
The Fibonacci Sequence and the Golden Ratio
We have previously touched upon Fibonacci sequences when discussing different types of …
The Problem with Solving Inequalities in GCSE Maths
In this blog, I will explain one of the most common mistakes …
5 things that make a good maths tutor
Following on from our previous blog on why university students make excellent …
Triangles in GCSE Maths
Triangles come up a lot in GCSE mathematics. And there are a …
A guide to trigonometry (SOHCAHTOA) – Part 2
In the last blog, I introduced a method for using trigonometry to …
A guide to trigonometry (SOHCAHTOA) – Part 1
Following on from the previous blog on Pythagoras’ theorem, this blog will …
Easy as Py – a guide to Pythagoras’ Theorem
a2 + b2 = c2.That’s Pythagoras’ theorem. What it says is that …

This site uses Akismet to reduce spam. Learn how your comment data is processed.